Related papers: Cuspidal edges on focal surfaces of regular surfac…

We study complex spatial quartic surfaces with simple singularities up to equisingular deformations; as a first step, give a complete equisingular deformation classification of the so-called non-special simple quartic surfaces.

In the paper, we investigate properties of the nine-dimensional variety of the inflection points of the plane cubic curves. The description of local monodromy groups of the set of inflection points near singular cubic curves is given. Also,…

Using the method of moving frames we analyze the algebra of differential invariants for surfaces in three-dimensional affine geometry. For elliptic, hyperbolic, and parabolic points, we show that if the algebra of differential invariants is…

In this paper, we consider deformations of singular complex curves on complex surfaces. Despite the fundamental nature of the problem, little seems to be known for curves on general surfaces. Let $C\subset S$ be a complete integral curve on…

We use a Simons type equation in order to characterize complete non-minimal pmc surfaces with non-negative Gaussian curvature.

The paper is devoted to differential geometric invariants determining a Frenet curve in up to a direct similarity These invariants can be presented by the Euclidean curvatures in terms of an arc lengths of the spherical indicatrices. Then,…

The main objective of the present paper is to investigate a sufficient condition for which a rectifying curve on a smooth surface remains invariant under isometry of surfaces, and also it is shown that under such an isometry the component…

We consider singular holomorphic foliations on compact complex surfaces with invariant rational nodal curve of positive self-intersection. Then, under some assumptions, we list all possible foliations.

We suggest an invariant way to enumerate nodal and nodal-cuspidal real deformations of real plane curve singularities. The key idea is to assign Welschinger signs to the counted deformations. Our invariants can be viewed as a local version…

We study singularities of constant positive Gaussian curvature surfaces and determine the way they bifurcate in generic 1-parameter families of such surfaces. We construct the bifurcations explicitly using loop group methods. Constant…

We prove that finite area isolated singularities of surfaces with constant positive curvature in R^3 are removable singularities, branch points or immersed conical singularities. We describe the space of immersed conical singularities of…

We investigate some characteristic properties of specific Weingarten surfaces in the three-dimensional Euclidean space using the nets of the lines of curvature resp. the asymptotic lines on both central surfaces of them.

In this study, we give the relationships between the conical curvatures of ruled surfaces drawn by the unit vectors of the ruling, central normal and central tangent of a regular ruled surface in the Euclidean -space. We obtain the…

The Eckardt hypersurface in $\mathbb{P}^{19}$ parameterizes smooth cubic surfaces with an Eckardt point, which is a point common to three of the $27$ lines on a smooth cubic surface. We describe the cubic surfaces lying on the singular…

We study the geometry of homogeneous hypersurfaces and their focal sets in complex hyperbolic spaces. In particular, we provide a characterization of the focal set in terms of its second fundamental form and determine the principal…

Invariant minimal surfaces in the real special linear group of degree 2 with canonical Riemannian and Lorentzian metrics are studied. Constant mean curvature surfaces with vertically harmonic Gau{\ss} map are classified.

We use PDE methods as developed for the Liouville equation to study the existence of conformal metrics with prescribed singularities on surfaces with boundary, the boundary condition being constant geodesic curvature. Our first result shows…

Here, we focus on focal surfaces of a tubular surface in Euclidean 3-space E^3: Firstly, we give the tubular surfaces with respect to Frenet and Darboux frames. Then, we define focal surfaces of these tubular surfaces. We get some results…

We study surfaces with constant anisotropic mean curvature which are invariant under a helicoidal motion. For functionals with axially symmetric Wulff shapes, we generalize the recently developed twizzler representation of Perdomo to the…

We study the Picard variety of the Fano surface of nodal and mildly cuspidal cubic threefolds in arbitrary characteristic by relating divisors on the Fano surface to divisors on the symmetric product of a curve of genus 4.